{
  "id": "18c7b74b-3cc8-5784-8ed0-62a8d9d6bee4",
  "slug": "duration",
  "term": "Duration",
  "aliases": [],
  "category": "Fixed Income",
  "category_slug": "fixed-income",
  "difficulty": "intermediate",
  "definition": "Duration is a measure of the sensitivity of a fixed income security's price to changes in interest rates, expressed as the weighted average time (in years) to receive the bond's cash flows, where Macaulay duration measures this time-weighted average and modified duration converts it into a direct price sensitivity measure—the percentage price change per 100 basis point change in yield.",
  "key_takeaways": [
    "Modified Duration ≈ –(ΔP/P) / Δy: a bond with modified duration of 7 will lose approximately 7% in price for a 100 bps rise in yield.",
    "Macaulay duration equals the weighted average time to cash flow receipt, with weights being each payment's present value divided by total bond price.",
    "Zero-coupon bonds have duration equal to their maturity; coupon bonds have duration less than maturity.",
    "Convexity is the second-order correction to duration; for large yield changes, duration understates price recovery in declining rate environments.",
    "Duration matching is the primary tool for immunizing bond portfolios against parallel yield curve shifts, used extensively by insurance companies and pension funds."
  ],
  "detailed_explanation": "Duration has evolved from a theoretical time-weighted measure to the central analytical framework for fixed income risk management. Frederick Macaulay first proposed the duration concept in 1938 as a more meaningful measure of a bond's 'length' than simple maturity, recognizing that a 10-year bond paying large annual coupons is economically shorter than a 10-year zero-coupon bond that pays everything at maturity.\n\nMacaulay Duration is calculated as: D_Mac = Σ [t × PV(CF_t)] / Bond Price, where t is the time period of each cash flow, PV(CF_t) is the present value of cash flow at time t. This weighted average tenure considers not just when principal is returned but when all intermediate coupons are received, weighted by their discounted value. A 5-year bond paying 6% semi-annual coupons at a 5% yield has a Macaulay duration of approximately 4.3 years—it behaves economically more like a 4.3-year zero-coupon bond than a 5-year bond.\n\nModified Duration converts Macaulay Duration into a direct price sensitivity measure: D_Mod = D_Mac / (1 + y/m), where y is the yield to maturity and m is the number of coupon periods per year. Modified duration gives the approximate percentage price change for a 1% (100 bps) change in yield: ΔP/P ≈ –D_Mod × Δy. For small yield changes, this linear approximation is accurate. For larger moves, the second-order term (convexity) must be added: ΔP/P ≈ –D_Mod × Δy + 0.5 × Convexity × (Δy)².\n\nDollar duration (DV01) translates modified duration into a dollar measure: DV01 = –(Modified Duration × Bond Price × 0.0001), representing the price change per basis point move in yield. DV01 is the primary risk unit used in fixed income portfolio management and hedging—traders express risk in terms of 'how many DV01 are you?' rather than abstract duration numbers.\n\nFor hedge funds running fixed income strategies—relative value, macro, carry, or credit—duration management is central to risk control. Interest rate duration is managed through government bond futures, interest rate swaps, or Treasury options. Spread duration (sensitivity to credit spread changes versus government yield changes) is managed separately through CDS, credit bond positions, and credit default swap indices. The separation of rate duration risk from credit spread risk enables fixed income traders to take pure credit views without inadvertent interest rate exposure.",
  "example": "A fixed income hedge fund holds $10 million face value of a 10-year U.S. Treasury bond with a 3.5% coupon, currently priced at 98.5 (YTM = 3.66%). The Macaulay duration is calculated as approximately 8.2 years. Modified Duration = 8.2 / (1 + 0.0366/2) = 8.2 / 1.0183 = 8.05. DV01 = 8.05 × $985,000 (price per $100K face × 100) × 0.0001 = $7,929 per basis point. If the Fed raises rates 25 bps unexpectedly, the approximate price change is: ΔP ≈ –8.05 × 0.0025 × $985,000 = –$19,823 per $100,000 face, or –$198,230 for the full $10 million position. The fund manager can hedge this risk by shorting Treasury futures: if the 10-year Treasury note futures contract has a DV01 of $900 per contract, the manager needs to short $7,929 / $900 ≈ 8.8 contracts (rounded to 9 contracts) to neutralize the duration risk.",
  "formula": "Modified Duration = Macaulay Duration / (1 + y/m); Price Change ≈ -Modified Duration × Δy × Price",
  "formula_latex": null,
  "interactive_type": "calculator",
  "calculator_id": null,
  "related_terms": [
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    "convexity",
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    "face-value",
    "futures-contract",
    "hedge-fund",
    "hedging",
    "interest-rate",
    "key-rate-duration",
    "macaulay-duration"
  ],
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    "federal-funds-rate",
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    "zero-coupon-bond",
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  "cross_references": [
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    "credit-default-swap",
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    "present-value",
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    "swap",
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  "tags": [
    "level:intermediate",
    "cat:fixed-income"
  ],
  "asset_classes": [
    "fixed-income"
  ],
  "regulators": [],
  "see_also": [],
  "sources": [
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  "wordcount": 709,
  "checksum": "dd13f6f65541dcac",
  "version": "2026.05.03",
  "license": "CC-BY-4.0",
  "updated_at": "2026-09-07T02:15:24+00:00",
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